Abstract

For a fixed singular modulus $\alpha$, we give an effective lower bound of the norm of $x-\alpha$ for another singular modulus $x$ with large discriminant. We then generalize this result for ${\Phi }_m\left(x,\alpha \right)$, where ${\Phi }_m\left(X,Y\right)\in \mathbb{Z}\left[X,Y\right]$ is the $m$-th modular polynomial.

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